Designing photonic structures of nanosphere arrays on reflectors for total absorptionE. Almpanis and N. Papanikolaou Citation: Journal of Applied Physics 114, 083106 (2013); doi: 10.1063/1.4818916 View online: http://dx.doi.org/10.1063/1.4818916 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/114/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High-Q silicon photonic crystal cavity for enhanced optical nonlinearities Appl. Phys. Lett. 105, 101101 (2014); 10.1063/1.4894441 Genetically designed L3 photonic crystal nanocavities with measured quality factor exceeding one million Appl. Phys. Lett. 104, 241101 (2014); 10.1063/1.4882860 The rigorous wave optics design of diffuse medium reflectors for photovoltaics J. Appl. Phys. 115, 153105 (2014); 10.1063/1.4872140 Electrotunable in-plane one-dimensional photonic structure based on silicon and liquid crystal Appl. Phys. Lett. 90, 011908 (2007); 10.1063/1.2430626 Silicon micromachined periodic structures for optical applications at λ = 1.55 μ m Appl. Phys. Lett. 89, 151110 (2006); 10.1063/1.2358323

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Designing photonic structures of nanosphere arrays on reflectorsfor total absorption

E. Almpanis and N. PapanikolaouDepartment of Microelectronics, IAMPPNM, NCSR “Demokritos,” GR-153 10 Athens, Greece

(Received 22 April 2013; accepted 5 August 2013; published online 23 August 2013)

By means of full electrodynamic simulations, we investigate structures that can totally absorb light

minimizing all reflections. Such efficient absorbers of visible and infrared light are useful in

photovoltaic and sensor applications. Our study provides a simple and transparent analysis of the

optical properties of structures comprising a resonant cavity and a reflector, which are the basic

ingredients of a resonant absorber, based on general principles of scattering theory. We concentrate

on periodic arrays of metallic or dielectric spherical particles in front of metallic or dielectric mirrors

and show that tuning the material absorption could turn resonances in the structures into total

absorption bands. Perfect absorption is predicted in metallic sphere arrays but also for Si spheres on a

metallic substrate, moreover, by replacing the substrate below the Si spheres with a lossless dielectric

Bragg mirror an all-dielectric-perfect-absorber is designed. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4818916]

I. INTRODUCTION

The development of smart surfaces that absorb electro-

magnetic (EM) waves has been driven by microwave and ra-

dar technology many years ago. Matching the impedance to

the environment of the incident radiation, using interfaces

with a graded index of refraction, is one way to reduce scat-

tering losses. Alternatively, different reflective layers can be

used to achieve destructive interference, and thus, reduce the

scattered radiation. Two of the oldest and simplest types of

absorbers are the Salisbury screens and the Dallenbach

layers.1 The Salisbury screen is a resonant absorber created

by placing a resistive sheet on a low-dielectric-constant

spacer in front of a metal plate. The Dallenbach layer con-

sists of a homogeneous lossy layer on top of a metallic plate.

Recently, there is renewed interest on surfaces that strongly

absorb visible and near-infrared light. Current interest is

mainly driven by energy harvesting and optical sensors

applications and aims at steering light in optically active

regions in a device as well as focusing in subwavelength

dimensions. Given the optical properties of the technologi-

cally available materials, current micro and nanofabrication

tools provide extra degrees of freedom for engineering

absorption at these wavelengths by patterning the surface in

the micro/nanoscale. In particular, thin discontinuous metal

films2 and nanostructured metallic surfaces exhibit a variety

of intriguing optical properties and allow for a variety of res-

onant absorbers designs. Subwavelength structures based on

metal-insulator-metal (MIM) involving Fabry-Perot-like

resonances are also often used to obtain enhanced absorp-

tion. Moreover, carefully designed metamaterial-based total

absorbers, which localize light strongly, were also

studied.3–21 Such effects are usually related to plasmonic

excitations and are very promising in the way of practical

applications in photovoltaic cells,22–25 visible and infrared

sensors,3,26 and detectors.27

The scattering properties of single plasmonic nanopar-

ticles have also drawn some attention recently. In particular,

metal-dielectric-metal meta-atoms with enhanced scattering

have been theoretically analyzed28 and systematic studies of

the absorption properties of spherical nanoparticles of differ-

ent metals have been reported.29,30

By generalizing the concept of critical coupling to a

nanocavity with surface plasmon resonances, it was pre-

dicted31 that coherent light can be completely absorbed and

transferred to surface plasmons in a metallic nanostructure.

The concept is general and can be understood as a time-

reversed process of the corresponding surface plasmon laser.

However, the creation of coherent light might be a constraint

for complex metallodielectric nanostructures. On the other

hand, the same principles are applicable to absorber designs

with resonators in front of a mirror. The presence of a reflec-

tor is useful since, by carefully designing the resonator, total

absorption condition could be possible for plane wave inci-

dent radiation. In the present work, we shall be concerned

with such perfect absorbers, consisting of periodic arrays of

micro and nanoparticles on top of a reflector.

Resonant absorbers have a limited spectral range. The

plasmon-based total absorbers show reduced linewidths,

which are beneficial for sensing applications based on refrac-

tive index change. Introducing structures that support multi-

ple resonances can increase the width of the absorption

band,3,6,7,18,32 which is useful in photovoltaics and thermo-

photovoltaics, but also in an attempt to replace traditional

absorbers with subwavelength thin metamaterial absorbers

for microwaves.19 Moreover, non-resonant, broadband

absorption using ultra-sharp convex metal grooves via adia-

batic nanofocusing of gap surface plasmon modes excited by

scattering off subwavelength-sized wedges was reported,33

while widening of the absorption spectra by combining two

MIM ribbons with different widths in the same subwave-

length period was also considered.18

0021-8979/2013/114(8)/083106/7/$30.00 VC 2013 AIP Publishing LLC114, 083106-1

JOURNAL OF APPLIED PHYSICS 114, 083106 (2013)

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A basic ingredient in many of the attempts to design per-

fect absorbers is the presence of a metallic material. Its role

is twofold: It localizes the EM energy in small subwave-

length volumes and at the same time the excited electrons of

the metal are responsible for absorbing light through Joule

effect. Recently, some attempts with dielectric absorbers

have also been reported.34,35 Moreover, total absorption in

an optically dense planar grid of plasmonic nanoparticles

over a transparent, high-index, dielectric substrate was also

investigated.36 One of the most successful and, due to its

simplicity, widely used designs is a periodic array of metallic

nanoparticles or stripes in close proximity to an optically

thick metallic film. The absorption band can be easily tuned

within the visible and infrared part of the spectrum by vary-

ing the dimensions of the structure. Symmetric nanoparticle

arrays are polarization insensitive, while for metallic stripes

or particles that brake the in-plane (x-y) symmetry absorp-

tion strongly depends on polarization, which provides more

design options, depending on the application. Moreover, in

many cases, absorption shows small dispersion with the inci-

dence angle, resulting in broad-angle absorbers.6,13,15,17,18

The condition for perfect absorption corresponds to the

zeros of the determinant of the scattering S matrix, which

connects the incoming to the outgoing field after scattering,

i.e., formally jout >¼ Sjin >. For a lossless structure, the

perfect absorption condition to have a vanishing outgoing

field leads to the trivial solution jin >¼ 0, since the condi-

tion detS¼ 0 is satisfied only for complex frequencies.

However, if losses are introduced, it is possible that some of

the zeros of the eigenvalues of the S matrix move to the real

axis and the perfect absorption condition is satisfied.37 This

is often called critical coupling condition,38 at which the in-

ternal, non-radiative loss rate cnr matches the radiation scat-

tering rate crad . The imaginary part of the dielectric function

(Ime), which determines material absorption, together with

the modal shape of the resonance are both critical parameters

and can be tailored to achieve total absorption using metals

or dielectrics.

Recent studies elaborated on the general case of a multi-

port resonator and derived the critical coupling condition

crad ¼ cnr in a general way.38 The concept was also demon-

strated theoretically for isolated metallic spheres and cylin-

ders,31 but in the latter case, the required input field must be

monochromatic and coherent. For spherical or cylindrical

structures, this means converging spherical or cylindrical

transverse magnetic (TM) waves, which can be hard to real-

ize experimentally.

In the present work we consider light absorption from

surfaces decorated with periodic arrays of spherical particles

in front of a reflector. The presence of a reflector offers a

wideband background for destructive interference, and we

consider an incoming plane wave for frequencies below the

first diffraction edge. High, and in most cases total, absorp-

tion is predicted for metallic or dielectric particles in front of

a metallic film or a dielectric Bragg mirror. After a short

description of our computational methodology, we first use a

generalized Drude model to describe a metallic nanoparticle

array in front of a metallic film and illustrate our analysis.

Then we demonstrate that almost total absorption can be

obtained with metals and dielectrics, described by experi-

mentally fitted optical constants, and present four different

geometries: (i) Ag spheres on a Ag film, (ii) Ag spheres on a

dielectric Bragg mirror, (iii) silicon spheres on a Ag film,

and (iv) an all dielectric structure with spheres on a Bragg

mirror. A theoretical analysis of the critical coupling effect

and the connection of the absorption and the local density of

states (LDOS) on the radiative and non radiative decay rates

is presented in the appendix.

II. COMPUTATIONAL METHOD

We use the layer-multiple-scattering (LMS) method to

study the optical properties of highly absorbing, periodically

patterned surfaces. The LMS method is a very powerful

computational tool for studying three dimensional (3D) pho-

tonic crystals consisting of non-overlapping spherical par-

ticles (scatterers) in a homogeneous host medium.39,40 The

method can directly provide the transmission, reflection, and

absorption coefficients of an incoming EM wave of any

polarization incident at a given angle on a finite slab of the

crystal and, therefore, it can describe an actual transmission

experiment. A further advantage of the method is that it does

not require periodicity in the direction perpendicular to the

layers, which must only have the same 2D periodicity. The

properties of the individual scatterers enter through the cor-

responding scattering T matrix which, for homogeneous

spherical particles, is given by the closed-form solutions of

the Mie-scattering problem. At a first step, in-plane multiple

scattering is evaluated in a spherical-wave basis using proper

propagator functions, where angular momentum expansion

up to ‘max ¼ 18 is sufficient to obtain convergence, for the

structures considered here. Subsequently, interlayer scatter-

ing is calculated in a plane-wave basis through appropriate

reflection and transmission matrices, where 81 plane waves

are typically used in the expansion. The scattering S matrix

of a multilayer slab, which transforms the incident into the

outgoing wave field, is obtained by combining the reflection

and transmission matrices of the individual component

layers in an appropriate manner. As a special case, a layer

can be a homogeneous plate or a planar interface between

two different homogeneous media. The reflection and trans-

mission matrices are then obtained directly by Fresnel equa-

tions and this possibility allows one to describe composite

structures which are used in an actual experiment and

include, e.g., a homogeneous spacer layer and/or a support-

ing substrate. The ratio of the transmitted or reflected energy

flux to the energy flux associated with the incident wave

defines the transmittance, T , or reflectance, R, of the slab,

while absorbance, A, is obtained through A ¼ 1�R� T .

III. RESULTS AND DISCUSSION

A. Metallic spheres on top of a metallic film

We start our discussion by considering an array of me-

tallic spheres of radius R¼ 70 nm arranged on a square lat-

tice with lattice constant a¼ 160 nm on top of an 100 nm,

optically thick, metallic film, separated by a 10 nm thick

SiO2 layer, in air, as shown in Fig. 1(a). Before presenting

083106-2 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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the results obtained with the realistic optical constants for

Ag, we use a modified Drude model to describe the metal

dielectric function

�met ¼ �1 �x2

P

x2 þ ixs�1: (1)

This function with �1 ¼ 3:718; �hxP ¼ 9:176 eV; �hs�1

¼ 0:021 eV can describe satisfactorily the optical response

of Ag in the visible, below 3 eV. The SiO2 dielectric function

was assumed constant: eSiO2¼ 2:13. This model is useful

since it allows the investigation of the influence of material

losses on the optical response in a systematic manner.

For normally incident light, the structure is highly re-

flective but has absorption resonances at the frequencies of

plasmon excitations. Structures with thinner metallic layers

have been also studied previously and show enhanced, extra-

ordinary, optical transmission.41 We focus our attention on a

particular mode around 2.96 eV. This mode derives from a

delocalized plasmon mode of the film and is characterized

by strong absorption and strong field enhancement in the

region where the spheres approach the metal film. The fre-

quency of the mode is sensitive to small changes in the thick-

ness of the SiO2 spacer. In Fig. 2(a), we show the change of

the absorption spectra for different values of the inverse

Drude relaxation time s�1, appearing in Eq. (1). As shown in

the appendix, the absorption has a Lorentzian form with the

half width at half maximum (HWHM) being equal to the

total decay rate ctot ¼ crad þ cnr. For the Drude dielectric

function of Eq. (1), absorption reaches values close to 60%.

By increasing s�1 by 4.6 times, we reach perfect, 100%,

absorption, but further increase reduces the peak maximum.

The absorption is accompanied by a strong field concentra-

tion in the region between the sphere and the film as shown

in Fig. 2(b). The radiative decay rate, crad, does not depend

on the absorption and can be extracted from the photonic

LDOS by fitting to Eq. (A2). This is calculated by neglecting

absorptive losses (s�1 ¼ 0) and is shown in Fig. 3(a). By

progressively increasing s�1, we increase cnr, and it is possi-

ble to determine ctot from the HWHM of the corresponding

absorption spectra. This is shown in Fig. 3(b) together with

the variation of the absorption maximum. It is worth to point

out that ctot determined from the absorption spectra,

extrapolated to s�1 ¼ 0, matches exactly crad determined

from LDOS. The critical coupling condition is satisfied

exactly and absorption reaches the maximum 100% when

ctot ¼ 2crad. For stronger losses, ctot does not vary linearly

with s�1, and also the fit of the absorption to a Lorentzian is

more difficult since the peaks are too broad and overlap

between neighboring resonances is strong. Alternatively,

using Eq. (A4) to fit the absorption spectrum, it is possible to

extract the total, the radiative, and the non-radiative decay

rates, from a single spectrum. The latter method is also reli-

able and reproduces the values extracted with the previous

procedures. The same approach can be used on experimental

spectra.

As already noted before,37 an arbitrary body or aggre-

gate can be made perfectly absorbing at discrete frequencies

if a precise amount of dissipation is added under specific

conditions of coherent monochromatic illumination.

Assuming linear dependence of ctot on the absorption con-

trolled by s�1, if the slope is known, we can predict the

amount of loss that is required to achieve perfect absorption.

Therefore, LDOS of the lossless structure can be used as a

guide for the design of efficient absorbers. Generally, struc-

tures that support sharp resonances, when losses are ignored,

are expected to become perfect absorbers if some low loss

comes into play. On the other hand, broader resonances

require materials with large imaginary part of the dielectric

function to yield perfect absorption.

FIG. 1. Schematic of the structures considered in this work. (a) Periodic

array of spheres on a metallic substrate with a thin dielectric spacer. (b)

Periodic array of spheres on top of a multilayer Bragg mirror.

FIG. 2. (a) Absorption spectra of light incident normally on a square array,

(a¼ 160 nm), of metallic spheres of radius 70 nm on top of a metallic film,

of thickness 100 nm separated by a 10 nm thick SiO2 layer as depicted in

Fig. 1(a). The calculations are carried out using the modified Drude dielec-

tric function of Eq. (1) (black line), and inverse relaxation time increased by

a factor of 4.6 (dashed line) and 10 (gray line). (b) Profile of the electric field

intensity, normalized to that of the incoming field, for the system with maxi-

mum peak absorption (dashed line of (a)) at the frequency of total

absorption.

083106-3 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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Perfect absorbers can be designed using periodic particle

arrays, since by varying the geometry, it is possible to fulfill

the critical coupling condition for a given material absorption.

To demonstrate this, we have chosen four different examples

using realistic optical constants of metals and dielectrics and

present structures where absorption is maximized.

In Fig. 4, we present the absorption spectra calculated

with the realistic optical parameters of Ag interpolated from

the tabulated data of Johnson and Christy,42 that will be used

from now on to describe Ag, for a structure shown in Fig.

1(a) with spheres of R¼ 65 nm, lattice constant a¼ 150 nm,

and a 3 nm thick SiO2 spacer layer on a 100 nm thick Ag

film. At normal incidence, total absorption is predicted close

to 2.68 eV while a somewhat smaller value is expected

around 3.18 eV. The lower frequency mode has a field profile

similar to the one shown in Fig. 2(b) with strong field con-

centration between the sphere and the metallic film. What is

interesting however is the angular dispersion of the modes.

In Figs. 4(b) and 4(c), we present the absorption for TM and

transverse electric (TE) polarized light incident at an angle

in the x-z plane. The higher frequency mode has a flat disper-

sion only for TE polarized light and shows a splitting for TM

polarization. The low frequency mode on the other hand has

almost flat dispersion with absorption close to one, even at

large angles of incidence, up to the light line, for both polar-

izations, i.e., this mode induces efficient broad-angle absorp-

tion. Modes with similar characteristics to the latter one are

reported in realizations of efficient absorbers involving parti-

cle arrays and a metallic film, and have been exploited to

achieve improved photovoltaic designs,24 and sensitive sen-

sors in visible and infrared frequencies.26

B. Silver spheres on a Bragg mirror

The metallic components are not essential to achieve

total absorption. For example, the metallic layer below the

metallic spheres acts mainly as a reflector and can be

replaced by a lossless dielectric Bragg mirror. To demon-

strate this, we have calculated the optical response of an

array of silver spheres on top of a TiO2/SiO2 Bragg mirror

which exhibits a band gap in the frequency region under con-

sideration. Similar to the previous structure, we consider an

array of Ag spheres with radius R¼ 65 nm arranged in a

square lattice with a¼ 150 nm on top of six periods of alter-

nated TiO2(45 nm)/SiO2(65 nm) layers, schematically shown

in Fig. 1(b), with eTiO2¼ 6:25; eSiO2

¼ 2:13. As shown in

Fig. 5(a), as the number of periods of the Bragg mirror

increases, and so does the reflectivity, the plasmonic reso-

nance of the metallic spheres at 3.15 eV is turned into a total

absorption band, while strong absorption also occurs around

3.4 eV. The electric field profile associated with the total

absorption resonance is very similar to the previous case of

the metallic mirror showing strong field localization in the

region where the spheres touch the flat surface. In this sys-

tem, absorption takes place in the metallic spheres only,

since the dielectrics are assumed to be lossless, but the analy-

sis with the decay rates evaluated from the appropriate reso-

nance widths still holds and leads to similar conclusions. The

role of the mirror here is limited to the creation of the

strongly localized resonant mode as a result of the interaction

with the plasmon resonance of the metallic spheres. The dis-

persion of the absorbing modes is shown in Figs. 5(b) and

5(c). For the lower frequency resonance, the dielectric mirror

causes a shift of the absorption band for TM polarization,

pushing the resonance to higher frequencies with increasing

angle of incidence. For TE polarization, the absorption band

is not shifted, but at large angles of incidence, as we

approach the light line, we observe the interaction with the

slab modes of the dielectric mirror. The higher frequency

FIG. 3. (a) Change of the density of photonic states for the structure of Fig.

1(a) with the same parameters as in Fig. 2 but using the modified Drude

dielectric function of Eq. (1) with s�1 ¼ 0. (b) Black line (left scale): ratio

of ctot=crad extracted by fitting the absorption spectra to Eq. (A4) versus the

inverse Drude relaxation time, and the dotted line is a linear interpolation.

The horizontal line marks the critical coupling condition. Gray line (right

scale): Maximum value of the corresponding absorption peak.

FIG. 4. (a) Absorption spectra of light incident normally on a square array

of lattice constant 150 nm of Ag spheres of radius 65 nm on top of a Ag film,

of thickness 100 nm separated by a 3 nm thick SiO2 spacer. Absorption spec-

tra for (b) TM and (c) TE polarized light incident at an angle in the xz-plane

versus the corresponding x component of the wave vector. The straight lines

mark the light line in air.

083106-4 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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resonance for TM polarization shows no dispersion, while

for TE polarization, there is a stronger signature of the Bragg

mirror slab modes.

Comparing these results with the case of Ag spheres on

a Ag film, we see that total absorption at normal incidence is

possible in both systems; however, the metallic film gives a

mode with flat dispersion, which is useful in applications

where broad-angle absorption is important.

C. Dielectric spheres on top of a metallic film

The spherical particles considered in the previous two

examples act as resonators, and metals could be replaced by

lossy dielectrics. To elucidate that, we consider a periodic

array of Si spheres on top of a 100 nm, optically thick, Ag

film separated by a 25 nm thick SiO2 spacer. The spheres

have a radius of 170 nm and are arranged on a square lattice

with lattice constant 360 nm. The absorption spectrum of this

system, using realistic optical constants for both silver42 and

silicon,43 is shown in Fig. 6(a). In the frequency range con-

sidered, just above the silicon electronic band gap, absorp-

tion is moderate, and we find two absorption peaks. The

lower frequency resonance is sharper, giving rise to almost

total absorption (99.7%) and it corresponds to a strong field

localization in the region between the spheres. The other res-

onance appears at higher frequency, it is broader, and the

associated field distribution is mostly localized in the region

where the spheres touch the substrate. This modal shape is

similar to that of the corresponding mode discussed previ-

ously in relation to the metallic spheres. The occurrence of a

frequency where perfect absorption takes place depends on

the interplay between material absorption and quality factor

of the resonance. Replacing metallic spheres with dielectric

ones results in sharper resonances and offers the possibility

to realize strong absorption with weakly absorbing materials.

It is to be noted that, despite the fact that larger radii are

required to support (Mie) resonances in dielectrics, as com-

pared to metals, the EM field can be strongly localized in

subwavelength volumes, as seen in the field profiles in Fig.

6. It is also interesting that losses occur mainly in silicon.

Our calculations show that ignoring the losses in silicon by

setting ImeSi ¼ 0 lowers the maximum absorption of the res-

onance at 1.38 eV to 8% and of the second resonance (at

1.61 eV) to 12%. We obtained similar results for low-refrac-

tive-index dielectric spheres, e.g., made of silica or of a poly-

mer material, on a metallic substrate. Experimental studies

of such structures with polystyrene44 and silica45 spheres on

a metallic film were recently reported and show multiple

sharp absorption peaks depending on the geometry.

However, due to the low losses of silica and polymers at visi-

ble and near infrared frequencies, we obtain absorption lower

than 85%, for realistic material parameters. The angle de-

pendence of the absorption is shown in Figs. 6(b) and 6(c).

For metallic particles above a metal film, discussed previ-

ously, absorption is related to plasmon excitation which has

a flat dispersion. For Si spheres, absorption of the low fre-

quency mode for TM polarized light persists with only a

small frequency shift, up to large angles of incidence, while

for TE polarized light, the absorption band shows strong dis-

persion, with the maximum dropping down and moving to

higher frequencies with increasing angle of incidence. The

behavior of the high frequency mode is more complex and

shows strong dispersion for both polarizations.

D. All-dielectric perfect absorbers

Following the previous analysis, metals can be avoided

and sharp resonances can lead to perfect absorption with low

loss materials, so as a final example, we consider high-re-

fractive-index dielectric spheres with Reesph ¼ 12 on a

FIG. 5. (a) The same as in Fig. 4 but with the Ag film and SiO2 spacer

replaced by a dielectric Bragg mirror consisting of six periods of alternated

TiO2(45 nm)/SiO2(65 nm) layers. The gray line in (a) refers to the array of

spheres deposited on top of 45 nm thick TiO2 film on a semi-infinite SiO2

substrate. Absorption spectra for (b) TM and (c) TE polarized light incident

at an angle in the xz-plane versus the corresponding x component of the

wave vector. The straight lines mark the light line in air. FIG. 6. (a) Absorption spectra of light incident normally on a square array,

of lattice constant 360 nm, of Si spheres, of radius 170 nm, on top of a

100 nm thick Ag film separated by a 25 nm thick SiO2 spacer, in air (see Fig.

1(b)). Absorption for TM (b) and TE (c) polarized light incident at an angle

in the x-z plane versus the corresponding x-component of the wavevector.

The strong lines mark the light line in air. (d) Electric field intensity profile,

normalized to that of the incoming field, associated with the low (d) and the

high (e) frequency resonances of Fig. 6(a).

083106-5 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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lossless Bragg mirror. Using dispersionless optical constants

allows us to present the results in scaled units. In Fig. 7, we

show the absorption spectra for different values of Imesph for

a periodic array of dielectric spheres of radius R¼ 0.8d,

where d is the multilayer periodicity (see Fig. 1), arranged

on a square lattice with a¼ 1.7d on top of a Bragg mirror,

consisting of 6 periods of TiO2ð0:4dÞ=SiO2ð0:6dÞ layers.

The spectrum is characterized by a series of sharp absorption

peaks, with their maxima strongly depending on the imagi-

nary part of the dielectric constant. We focus on two charac-

teristic resonance peaks. The first one close to x1 ¼ 0:42a=kgives almost total absorption (99%) for moderate

Imesphð¼ 0:02Þ. The associated field intensity pattern shows

strong enhancement in the region where the spheres

approach each other in the lattice. The second resonance, at

x2 ¼ 0:52a=k, corresponds to the strongest field, which is

mainly concentrated in the region where the spheres touch

the upper TiO2 layer of the Bragg mirror. A mode with simi-

lar field profile is found in all configurations we studied pre-

viously and is expected to give a strongly absorbing band. In

Fig. 7(b), we plot the variation of ctot=crad for the two

absorption resonances, at x1 and x2, versus Imesph. We

observe a linear increase with a different slope for each reso-

nance, which depends on the modal shape as well as on the

absorption of the different materials in the structure. In any

case, it is possible to predict the exact amount of losses

required to reach the critical coupling condition and achieve

perfect absorption. Such calculations can help the design of

total absorbers if the dielectric material losses can be tai-

lored, for example, by chemical modification, controlling the

surface roughness, or even by inserting a thin highly absorp-

tive layer in the regions of high field concentration.

Another interesting aspect of this structure is the angular

dispersion of the absorption. This is shown in Fig. 8. There

are many similarities of these spectra to those obtained for

the array of Si spheres on top of the Ag mirror (see Fig. 6).

The low frequency resonance x1 shows only a weak angle

dependence for the TM polarization. However, the TE spec-

tra are dominated by the multilayer eigenmodes which

appear as a set of six modes almost parallel to each other.

These are slab modes of the TiO2/SiO2 multilayer, but due to

the presence of the periodic array can now leak outside. The

coupling of these modes to the sphere array modes of the

same symmetry will finally lead to hybridization minigaps

when modes cross.

IV. CONCLUSIONS

We have investigated visible and infrared resonant

absorbers and proposed different possibilities to achieve per-

fect absorption with metals and dielectrics. We offered a

simple and efficient quantitative analysis of resonant absorp-

tion based on scattering theory, which can be also used for

the design of simple subwavelength optical coatings demon-

strated recently46,47 or metamaterial perfect absorbers. Our

results would be useful in the design of perfect absorbers

using particle arrays in front of reflectors since particles of

different shapes like nanorods, nanorings, etc., could be sim-

ilarly optimized to achieve perfect light absorbers.

ACKNOWLEDGMENTS

E. Almpanis was supported by NCSR “Demokritos”

through a postgraduate fellowship.

APPENDIX ABSORPTION OF A LOSSY RESONATORIN FRONT OF A REFLECTOR

We follow the appendix of Gantzounis and Stefanou48

and proceed further by considering dissipative losses. Our

approach relies on general properties of the scattering matrix

S implied from the causality condition.37,49 For a lossless

FIG. 7. (a) Absorption spectra of light incident normally on a square array

of dielectric spheres on top of a TiO2/SiO2 Bragg mirror for different values

of the dielectric constant of the spheres. esph ¼ 12þ i0:02: black full line,

esph ¼ 12þ i0:05: red dashed line, esph ¼ 12þ i0:10: green dashed-dotted

line. (b) Ratio of the total to the radiative decay rates for two of the resonan-

ces shown in (a), x1: full line, x2: dashed line. The horizontal line marks

the critical coupling condition. (c) Electric field profile, normalized to that

of the incoming field, for spheres with esph ¼ 12þ i0:02 at x1. (d) Field pro-

file for spheres with esph ¼ 12þ i0:02 at x2.

FIG. 8. Absorption spectra for a square array of dielectric spheres

(esph ¼ 12þ i0:02) on top of a lossless TiO2/SiO2 Bragg mirror for (a) TM

and (b) TE polarized light incident at an angle in the xz-plane versus the cor-

responding x component of the wave vector. The straight lines mark the light

line in air.

083106-6 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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structure comprising a planar total reflector, the S matrix is

unitary and, since transmission vanishes, its eigenvalues are

reduced to the appropriate reflection coefficients which have

the form48

rðxÞ ¼ e2i/ x� xn � icrad

x� xn þ icrad

(A1)

about the resonance frequencies xn. The analytic continua-

tion of these eigenvalues implies zeros in the upper complex

plane at xn þ icrad and poles in the lower half at xn � icrad .

From this form, it is easy to obtain the local density of pho-

tonic states in the vicinity of xn48

DnðxÞ � 1

pcrad

ðx� xnÞ2 þ crad2: (A2)

The distance of a pole (zero) from the real axis is directly

related to the half-width at half maximum of the resonance

and crad is the associated decay rate. If losses are introduced

in the system, then the S matrix is no longer unitary and the

poles and zeros of its eigenvalues are shifted down in the

complex frequency plane by an additional amount �icnr.37

Therefore in the dissipative system, we have

rðxÞ ¼ e2i/ x� xn � iðcrad � cnrÞx� xn þ iðcrad þ cnrÞ

; (A3)

where the phase / becomes complex: / ¼ hþ ia. In this

case, since transmittance is zero, the absorbance in the vicin-

ity of a resonance is given by

A ¼ 1� jrðxÞj2 ¼ 1� e�4a þ 4e�4acradcnr

ðx� xnÞ2 þ ðcrad þ cnrÞ2:

(A4)

This equation includes losses away from the resonant fre-

quency and is identical to Eq. (11) of Ref. 38 if a ¼ 0.

Assuming that a is frequency independent within the reso-

nance bandwidth, Eq. (A4) allows for the determination of

crad; cnr, and a by fitting the relevant absorption spectra. We

note that crad can also be extracted independently from the

local density of states calculated for the corresponding loss-

less structure.

1E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed.

(Scitech Publication, Raleigh, 2006).2N. Stefanou and A. Modinos, J. Phys. Condens. Matter 3, 8149 (1991).3M. G. Nielsen, A. Pors, O. Albrektsen, and S. I. Bozhevolnyi, Opt.

Express 20, 13311 (2012).4H. Y. Zheng, X. R. Jin, J. W. Park, Y. H. Lu, J. Y. Rhee, W. H. Jang, and

Y. P. Lee, Opt. Express 20, 24002 (2012).5P. Ding, E. Liang, G. Cai, W. Hu, C. Fan, and Q. Xue, J. Opt. 13, 075005

(2011).6Y. Cui, K. H. Fung, J. Xu, S. He, and N. X. Fang, Opt. Express 20, 17552

(2012).7X. Shen, T. J. Cui, J. Zhao, H. F. Ma, W. X. Jiang, and H. Li, Opt. Express

19, 9401 (2011).8A. Polyakov, S. Cabrini, S. Dhuey, B. Harteneck, P. J. Schuck, and H. A.

Padmore, Appl. Phys. Lett. 98, 203104 (2011).

9N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, Phys.

Rev. Lett. 100, 207402 (2008).10J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, and M. Qiu, Appl. Phys.

Lett. 96, 251104 (2010).11H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J.

Padilla, Opt. Express 16, 7181 (2008).12J. Hao, L. Zhou, and M. Qiu, Phys. Rev. B 83, 165107 (2011).13Z. Fang, Y.-R. Zhen, L. Fan, X. Zhu, and P. Nordlander, Phys. Rev. B 85,

245401 (2012).14C. Wu, Y. Avitzour, G. Shvets, M. A. Noginov, and N. I. Zheludev, Proc.

SPIE 7029, 70290W (2008).15Y. Avitzour, Y. A. Urzhumov, and G. Shvets, Phys. Rev. B 79, 045131

(2009).16J. Zhang, J.-Y. Ou, K. F. MacDonald, and N. I. Zheludev, J. Opt. 14,

114002 (2012).17K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, Nat. Commun. 2,

517 (2011).18P. Bouchon, C. Koechlin, F. Pardo, R. Ha€ıdar, and J.-L. Pelouard, Opt.

Lett. 37, 1038 (2012).19H. Li, L. H. Yuan, B. Zhou, X. P. Shen, Q. Cheng, and T. J. Cui, J. Appl.

Phys. 110, 014909 (2011).20D. Y. Shchegolkov, A. K. Azad, J. F. OHara, and E. I. Simakov, Phys.

Rev. B 82, 205117 (2010).21C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).22C. H€agglund, S. P. Apell, and B. Kasemo, Nano Lett. 10, 3135 (2010).23H. A. Atwater and A. Polman, Nature Mater. 9, 205 (2010).24N. C. Lindquist, W. A. Luhman, S. H. Oh, and R. J. Holmes, Appl. Phys.

Lett. 93, 123308 (2008).25J. N. Munday and H. A. Atwater, Nano Lett. 11, 2195 (2011).26N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, Nano Lett. 10,

2342 (2010).27J. J. Talghader, A. S. Gawarikar, and R. P. Shea, Light: Sci. Appl. 1, e24

(2012).28Z. Ruan and S. Fan, Phys. Rev. Lett. 105, 013901 (2010).29B. S. Lukyanchuk, A. E. Miroshnichenko, M. I. Tribelsky, Y. S. Kivshar,

and A. R. Khokhlov, New J. Phys. 14, 093022 (2012).30M. I. Tribelsky, Europhys. Lett. 94, 14004 (2011).31H. Noh, Y. Chong, A. D. Stone, and H. Cao, Phys. Rev. Lett. 108, 186805

(2012).32C. Tserkezis, G. Gantzounis, and N. Stefanou, J. Phys. Condens. Matter

20, 075232 (2008).33T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J.

Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, Nat. Commun. 3,

969 (2012).34P. Spinelli, M. A. Verschuuren, and A. Polman, Nat. Commun. 3, 692

(2012).35A. Raman, Z. Yu, and S. Fan, Opt. Express 19, 19015 (2011).36M. Albooyeh and C. R. Simovski, Opt. Express 20, 21888 (2012).37Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, Phys. Rev. Lett. 105,

053901 (2010).38J. Yoon, K. H. Seol, S. H. Song, and R. Magnusson, Opt. Express 18,

25702 (2010).39N. Stefanou, V. Yannopapas, and A. Modinos, Comput. Phys. Commun.

113, 49 (1998).40N. Stefanou, V. Yannopapas, and A. Modinos, Comput. Phys. Commun.

132, 189 (2000).41N. Papanikolaou, Phys. Rev. B 75, 235426 (2007).42B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).43D. E. Aspnes, in Properties of Silicon, EMIS Data Reviews (Inspec IEE,

London, 1988).44M. L�opez-Garc�ıa, J. F. Galisteo-L�opez, C. L�opez, and A. Garc�ıa-Mart�ın,

Phys. Rev. B 85, 235145 (2012).45J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, Adv.

Mater. 23, 1272 (2011).46M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, Nature Mater. 12,

20 (2013).47M. A. Kats, D. Sharma, J. Lin, P. Genevet, R. Blanchard, Z. Yang, M. M.

Qazilbash, D. N. Basov, S. Ramanathan, and F. Capasso, Appl. Phys. Lett.

101, 221101 (2012).48G. Gantzounis and N. Stefanou, Phys. Rev. B 72, 075107 (2005).49N. G. van Kampen, Phys. Rev. 89, 1072 (1953).

083106-7 E. Almpanis and N. Papanikolaou J. Appl. Phys. 114, 083106 (2013)

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